TY - JOUR

T1 - Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models

AU - Hara, T.

AU - Hofstad, van der, R.W.

AU - Slade, G.

PY - 2003

Y1 - 2003

N2 - We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ${\mathbb{Z}^d}$, having long finite-range connections, above their upper critical dimensions $d=4$ (self-avoiding walk), $d=6$ (percolation) and $d=8$ (trees and animals). The two-point functions for these models are respectively the generating function for self-avoiding walks from the origin to $x \in {\mathbb{Z}^d}$, the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of $|x|^{2-d}$ as $x \to \infty$. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoiding walk, the triangle condition for percolation, and the square condition for lattice trees and lattice animals, for sufficiently spread-out models above the upper critical dimension.

AB - We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ${\mathbb{Z}^d}$, having long finite-range connections, above their upper critical dimensions $d=4$ (self-avoiding walk), $d=6$ (percolation) and $d=8$ (trees and animals). The two-point functions for these models are respectively the generating function for self-avoiding walks from the origin to $x \in {\mathbb{Z}^d}$, the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of $|x|^{2-d}$ as $x \to \infty$. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoiding walk, the triangle condition for percolation, and the square condition for lattice trees and lattice animals, for sufficiently spread-out models above the upper critical dimension.

U2 - 10.1214/aop/1046294314

DO - 10.1214/aop/1046294314

M3 - Article

VL - 31

SP - 349

EP - 408

JO - The Annals of Probability

JF - The Annals of Probability

SN - 0091-1798

IS - 1

ER -